Topological Dynamics: Minimality, Entropy and Chaos.

Sergiy Kolyada

Institute of Mathematics, NAS of Ukraine, Kyiv

Zentrum Mathematik, Technische Universit¨at Munchen,¨ John-von-Neumann Lecture, 2013

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. Throughout this part of the lecture (X , f ) denotes a topological dynamical system, where X is a (compact) metric space with a metric d and f : X → X is a continuous map. A point x ∈ X ’moves’, its trajectory being the sequence x, f (x), f 2(x), f 3(x),... , where f n is the n-th iteration of f . The point f n(x) is the position of x after n units of time. The set of points of the trajectory of x under f is called the orbit of x, denoted by orbf (x). Topological transitivity is a global characteristic of a dynamical system. The concept of topological transitivity goes back to G. D. Birkhoff, he used him in 1920.

1.Transitive maps 1. Topologically Transitive Maps Introduction

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. A point x ∈ X ’moves’, its trajectory being the sequence x, f (x), f 2(x), f 3(x),... , where f n is the n-th iteration of f . The point f n(x) is the position of x after n units of time. The set of points of the trajectory of x under f is called the orbit of x, denoted by orbf (x). Topological transitivity is a global characteristic of a dynamical system. The concept of topological transitivity goes back to G. D. Birkhoff, he used him in 1920.

1.Transitive maps 1. Topologically Transitive Maps Introduction

Throughout this part of the lecture (X , f ) denotes a topological dynamical system, where X is a (compact) metric space with a metric d and f : X → X is a continuous map.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. Topological transitivity is a global characteristic of a dynamical system. The concept of topological transitivity goes back to G. D. Birkhoff, he used him in 1920.

1.Transitive maps 1. Topologically Transitive Maps Introduction

Throughout this part of the lecture (X , f ) denotes a topological dynamical system, where X is a (compact) metric space with a metric d and f : X → X is a continuous map. A point x ∈ X ’moves’, its trajectory being the sequence x, f (x), f 2(x), f 3(x),... , where f n is the n-th iteration of f . The point f n(x) is the position of x after n units of time. The set of points of the trajectory of x under f is called the orbit of x, denoted by orbf (x).

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps Introduction

Throughout this part of the lecture (X , f ) denotes a topological dynamical system, where X is a (compact) metric space with a metric d and f : X → X is a continuous map. A point x ∈ X ’moves’, its trajectory being the sequence x, f (x), f 2(x), f 3(x),... , where f n is the n-th iteration of f . The point f n(x) is the position of x after n units of time. The set of points of the trajectory of x under f is called the orbit of x, denoted by orbf (x). Topological transitivity is a global characteristic of a dynamical system. The concept of topological transitivity goes back to G. D. Birkhoff, he used him in 1920.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps Introduction

Intuitively, a topologically transitive map f has points which eventually move under iteration from one arbitrarily small neighbourhood to any other. Consequently, the dynamical system cannot be broken down or decomposed into two subsystems (disjoint sets with nonempty interiors) which do not interact under f , i.e., are invariant under the map (A ⊂ X is invariant if f (A) ⊂ A).

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. As usual, we adopt the condition (TT) as the definition of topological transitivity, but note that some authors take (DO) instead.

1.Transitive maps 1. Topologically Transitive Maps Introduction

Let X be a metric space and f : X → X continuous. Consider the following two conditions: (TT) for every pair of nonempty open (opene) sets U and V in X , there is a positive integer n such that f n(U) ∩ V 6= ∅,

(DO) there is a point x0 ∈ X such that the orbit of x0 is dense in X .

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps Introduction

Let X be a metric space and f : X → X continuous. Consider the following two conditions: (TT) for every pair of nonempty open (opene) sets U and V in X , there is a positive integer n such that f n(U) ∩ V 6= ∅,

(DO) there is a point x0 ∈ X such that the orbit of x0 is dense in X .

As usual, we adopt the condition (TT) as the definition of topological transitivity, but note that some authors take (DO) instead.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps Introduction

Any point with dense orbit is called a transitive point. A point which is not transitive is called intransitive. The set of transitive or intransitive points of (X , f ) will be denoted by tr(f ) or intr(f ) respectively, provided no misunderstanding can arise concerning the phase space. In such a case we will also speak on transitive or intransitive points of f .

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps On the equivalent formulations of the definition

The two conditions (TT) and (TT) are independent in general. In fact, 1 take X = {0} ∪ { n : n ∈ N} endowed with the usual metric and 1 1 f : X → X defined by f (0) = 0 and f ( n ) = n+1 , n = 1, 2,... . Clearly, f is continuous. The point x0 = 1 is (the only) transitive point for (X , f ) 1 but the system is not topologically transitive (take, say, U = { 2 }, V = {1}). So, (DO) does not imply (TT).

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps On the equivalent formulations of the definition

We show that neither (TT) implies (DO). To this end take I = [0, 1] and the standard tent map g(x) = 1 − |2x − 1| from I to itself. Let X be the set of all periodic points of g and f = g|X (a point x is periodic for g if g n(x) = x for some positive integer n; the least such n is called the period of x). Then the system (X , f ) does not satisfy the condition (DO), since X is infinite (dense in I ) while the orbit of any periodic point is finite. But the condition (TT) is fulfilled. This follows from the fact that for any nondegenerate subinterval J of I there is a positive integer k k with g (J) = I . Hence, whenever J1 and J2 are nonempty open subintervals of I , there is a periodic orbit of g which intersects both J1 and J2. This gives (TT) for (X , f ).

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps On the equivalent formulations of the definition

Nevertheless, under some additional assumptions on the phase space (or on the map) the two definitions (TT) and (DO) are equivalent. In fact, we have the following Silverman theorem: Theorem 1.1 If X has no isolated point then (DO) implies (TT). If X is separable and second category then (TT) implies (DO).

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps On the equivalent formulations of the definition

Beweis.

1. If X has no isolated point and {x0, x1, ...} is a dense orbit (where n xn := f (x0), n ≥ 0), then any opene sets U and V (in X ) there are xk ∈ U and xm ∈ V \{x0, x1, ..., xk }. But if m > k then f m−k (U) ∩ V 6= ∅.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. It is known that a metric space is separable iff it is second countable (means has a countable base).

1.Transitive maps 1. Topologically Transitive Maps On the equivalent formulations of the definition

Let’s recall some definitions. Definitions A set A ⊆ X is the first category (sometimes it’s called meager) if it the union of countably many nowhere dense subset of X . A set B ⊆ X is the second category if it is not the first one. A space X is called separable if it contains a countable dense subset.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps On the equivalent formulations of the definition

Let’s recall some definitions. Definitions A set A ⊆ X is the first category (sometimes it’s called meager) if it the union of countably many nowhere dense subset of X . A set B ⊆ X is the second category if it is not the first one. A space X is called separable if it contains a countable dense subset.

It is known that a metric space is separable iff it is second countable (means has a countable base).

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. Let us also remark that in general compact metric spaces, (TT) and (DO) are equivalent for onto maps. If f is transitive (i.e.,it satisfies (TT)) then f is onto. If a compact metric space admits a transitive map (i.e., if there exists a continuous selfmap f of X satisfying (TT)) then X has no isolated point if and only if it is infinite.

1.Transitive maps 1. Topologically Transitive Maps On the equivalent formulations of the definition

Beweis (Cont.) ∞ 2. So, let f has no a dense orbit, and let {Vn}n=1 be a countable base. k Therefore for any point x ∈ X there exists Vn(x) such that f (x) 6∈ Vn(x) ∞ −k for any k ≥ 0. But ∪k=0f (Vn(x)) is opene and intersects with any opene subset of X , because (TT). ∞ −k Let An(x) := X \ ∪k=0f (Vn(x)). Then An(x) contains x and is a closed nowhere dense subset of X . But in this case X = ∪x∈X An(x) is the union of a countably many nowhere dense subsets of X . What is a contradiction with X - the second category.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps On the equivalent formulations of the definition

Beweis (Cont.) ∞ 2. So, let f has no a dense orbit, and let {Vn}n=1 be a countable base. k Therefore for any point x ∈ X there exists Vn(x) such that f (x) 6∈ Vn(x) ∞ −k for any k ≥ 0. But ∪k=0f (Vn(x)) is opene and intersects with any opene subset of X , because (TT). ∞ −k Let An(x) := X \ ∪k=0f (Vn(x)). Then An(x) contains x and is a closed nowhere dense subset of X . But in this case X = ∪x∈X An(x) is the union of a countably many nowhere dense subsets of X . What is a contradiction with X - the second category.

Let us also remark that in general compact metric spaces, (TT) and (DO) are equivalent for onto maps. If f is transitive (i.e.,it satisfies (TT)) then f is onto. If a compact metric space admits a transitive map (i.e., if there exists a continuous selfmap f of X satisfying (TT)) then X has no isolated point if and only if it is infinite.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. A point x ∈ X is called nonwandering if for every neighbourhood U of x there is a positive integer n such that f n(U) ∩ U 6= ∅. The set of all nonwandering points of f will be denoted by Ω(f ).

1.Transitive maps 1. Topologically Transitive Maps Other equivalent definitions

More definitions. Definitions Given a dynamical system (X , f ), the ω-limit set of a point x ∈ X under f , ωf (x), is the set of all limit points of the trajectory of x, i.e., y ∈ ωf (x) n if and only if f k (x) → y for some sequence of integers nk → ∞.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps Other equivalent definitions

More definitions. Definitions Given a dynamical system (X , f ), the ω-limit set of a point x ∈ X under f , ωf (x), is the set of all limit points of the trajectory of x, i.e., y ∈ ωf (x) n if and only if f k (x) → y for some sequence of integers nk → ∞. A point x ∈ X is called nonwandering if for every neighbourhood U of x there is a positive integer n such that f n(U) ∩ U 6= ∅. The set of all nonwandering points of f will be denoted by Ω(f ).

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps Other equivalent definitions

The next result follows easily from the definitions and parts of it can be found in any book dealing at least partially with topological dynamics:

1 E. Akin, The general topology of dynamical systems, Graduate studies in mathematics, vol. 1, Amer. Math. Soc., 1993. 2 J. de Vries, Elements of topological dynamics, Mathematics and its applications vol. 257, Kluwer, Dordrecht, 1993.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps Other equivalent definitions

Theorem 1.2 Let (X , f ) be a dynamical system. Then the following are equivalent: 1. f is topologically transitive (i.e., (TT) is fulfilled), 2. for every pair of opene sets U and V in X , there is a nonnegative integer n such that f n(U) ∩ V 6= ∅, S∞ n 3. for every opene set U in X , n=1 f (U) is dense in X , S∞ n 4. for every opene set U in X , n=0 f (U) is dense in X , 5. for every pair of opene sets U and V in X , there is a positive integer n such that f −n(U) ∩ V 6= ∅, 6. for every pair of opene sets U and V in X , there is a nonnegative integer n such that f −n(U) ∩ V 6= ∅, S∞ −n 7. for every opene set U in X , n=1 f (U) is dense in X , S∞ −n 8. for every opene set U in X , n=0 f (U) is dense in X ,

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps Other equivalent definitions

Theorem 1.2 (Cont.) 9. if E ⊂ X is closed and f (E) ⊂ E then E = X or E is nowhere dense in X , 10. if U ⊂ X is open and f −1(U) ⊂ U then U = ∅ or U is dense in X ,

11. there exists a point x ∈ X such that ωf (x) = X,

12. there exists a Gδ-dense set A ⊂ X such that ωf (x) = X whenever x ∈ A,

13. the set tr(f ) is Gδ-dense, 14. the map f is onto and the set tr(f ) is nonempty, 15.Ω( f ) = X and tr(f ) is nonempty,

16. there is a point x ∈ X such that the set orbf (x0) \{x0} is dense in X.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps Other equivalent definitions

Theorem 1.2 (Cont.) Further, the above conditions imply that (17) the set tr(f ) is nonempty (i.e., (DO) is fulfilled). If, additionally, (X , f ) is without isolated points then also (17) is equivalent to the rest.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. If h is not a homeomorphism but only a continuous surjection (a semiconjugacy), then the transitivity of f implies the transitivity of g but not conversely.

1.Transitive maps 1. Topologically Transitive Maps Topological transitivity and conjugacy

Note also that topological transitivity is preserved by topological conjugacy. More precisely, let (X , f ) and (Y , g) be two dynamical systems and suppose they are topologically conjugate, i.e., there is a homeomorphism h : X → Y such that h ◦ f = g ◦ h. Then f is topologically transitive on X if and only if g is topologically transitive on Y .

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps Topological transitivity and conjugacy

Note also that topological transitivity is preserved by topological conjugacy. More precisely, let (X , f ) and (Y , g) be two dynamical systems and suppose they are topologically conjugate, i.e., there is a homeomorphism h : X → Y such that h ◦ f = g ◦ h. Then f is topologically transitive on X if and only if g is topologically transitive on Y . If h is not a homeomorphism but only a continuous surjection (a semiconjugacy), then the transitivity of f implies the transitivity of g but not conversely.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. Perhaps the most popular and/or most simple examples of transitive systems are the following ones. Example 1.1

Let (X , f ) be any dynamical system and let x0 ∈ X be a periodic point of f . Denote the (finite) orbit of x0 by Y and let g = f|Y . Then the dynamical system (Y , g) is transitive. Though it has isolated points, it satisfies both (TT) and (DO), hence all 17 conditions from Theorem 1.2. Notice also that (Y , g) is minimal.

Example 1.2 Let S be the unit circle and f : S → S be an irrational rotation. Then (S, f ) is topologically transitive, in fact minimal.

1.Transitive maps 1. Topologically Transitive Maps Some examples

First recall that a dynamical system is called minimal if all points are transitive. Trivially, minimality of a dynamical system implies its transitivity but not conversely.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. Example 1.2 Let S be the unit circle and f : S → S be an irrational rotation. Then (S, f ) is topologically transitive, in fact minimal.

1.Transitive maps 1. Topologically Transitive Maps Some examples

First recall that a dynamical system is called minimal if all points are transitive. Trivially, minimality of a dynamical system implies its transitivity but not conversely. Perhaps the most popular and/or most simple examples of transitive systems are the following ones. Example 1.1

Let (X , f ) be any dynamical system and let x0 ∈ X be a periodic point of f . Denote the (finite) orbit of x0 by Y and let g = f|Y . Then the dynamical system (Y , g) is transitive. Though it has isolated points, it satisfies both (TT) and (DO), hence all 17 conditions from Theorem 1.2. Notice also that (Y , g) is minimal.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps Some examples

First recall that a dynamical system is called minimal if all points are transitive. Trivially, minimality of a dynamical system implies its transitivity but not conversely. Perhaps the most popular and/or most simple examples of transitive systems are the following ones. Example 1.1

Let (X , f ) be any dynamical system and let x0 ∈ X be a periodic point of f . Denote the (finite) orbit of x0 by Y and let g = f|Y . Then the dynamical system (Y , g) is transitive. Though it has isolated points, it satisfies both (TT) and (DO), hence all 17 conditions from Theorem 1.2. Notice also that (Y , g) is minimal.

Example 1.2 Let S be the unit circle and f : S → S be an irrational rotation. Then (S, f ) is topologically transitive, in fact minimal.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. If J is a closed subinterval of I which does not 1 k 1 contain 2 then f (J) is twice as long as J. Therefore f (J) contains 2 for some k. Then f k+2(J) is a closed interval containing 0 and repeating the argument with doubling length we get that f n(J) = I for some n. This property easily implies transitivity (in general the property is stronger than transitivity and is called topological exactness). The system is not minimal, since the set of periodic points is dense in I . (No minimal systems exist on I .)

1.Transitive maps 1. Topologically Transitive Maps Some examples

Example 1.3 Let I = [0, 1] and f ∈ C(I ) be the standard tent map defined by f (x) = 1 − |2x − 1|.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. This property easily implies transitivity (in general the property is stronger than transitivity and is called topological exactness). The system is not minimal, since the set of periodic points is dense in I . (No minimal systems exist on I .)

1.Transitive maps 1. Topologically Transitive Maps Some examples

Example 1.3 Let I = [0, 1] and f ∈ C(I ) be the standard tent map defined by f (x) = 1 − |2x − 1|. If J is a closed subinterval of I which does not 1 k 1 contain 2 then f (J) is twice as long as J. Therefore f (J) contains 2 for some k. Then f k+2(J) is a closed interval containing 0 and repeating the argument with doubling length we get that f n(J) = I for some n.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps Some examples

Example 1.3 Let I = [0, 1] and f ∈ C(I ) be the standard tent map defined by f (x) = 1 − |2x − 1|. If J is a closed subinterval of I which does not 1 k 1 contain 2 then f (J) is twice as long as J. Therefore f (J) contains 2 for some k. Then f k+2(J) is a closed interval containing 0 and repeating the argument with doubling length we get that f n(J) = I for some n. This property easily implies transitivity (in general the property is stronger than transitivity and is called topological exactness). The system is not minimal, since the set of periodic points is dense in I . (No minimal systems exist on I .)

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. Further, if a dynamical system (X , f ) is transitive then by Theorem 1.2 (1. ⇒ 13.) it has a Gδ-dense set of transitive points. We show how this can be proved (provided 1. ⇒ 8. has been established). Realize that a point is transitive if and only if it visits all sets from a ∞ countable base {Ui }i=1 of open sets (note that X is compact). Thus

∞ ∞ ! \ [ −n tr(f ) = f (Uk ) . k=1 n=0 Now it is sufficient to use 1. ⇒ 8. in Theorem 1.2.

1.Transitive maps 1. Topologically Transitive Maps Transitive and intransitive points

First realize that in a dynamical system (X , f ), x ∈ intr(f ) ⇒ f (x) ∈ intr(f ) and f (x) ∈ tr(f ) ⇒ x ∈ tr(f ). If the system is without isolated points then we have equivalences instead of implications.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. We show how this can be proved (provided 1. ⇒ 8. has been established). Realize that a point is transitive if and only if it visits all sets from a ∞ countable base {Ui }i=1 of open sets (note that X is compact). Thus

∞ ∞ ! \ [ −n tr(f ) = f (Uk ) . k=1 n=0 Now it is sufficient to use 1. ⇒ 8. in Theorem 1.2.

1.Transitive maps 1. Topologically Transitive Maps Transitive and intransitive points

First realize that in a dynamical system (X , f ), x ∈ intr(f ) ⇒ f (x) ∈ intr(f ) and f (x) ∈ tr(f ) ⇒ x ∈ tr(f ). If the system is without isolated points then we have equivalences instead of implications. Further, if a dynamical system (X , f ) is transitive then by Theorem 1.2 (1. ⇒ 13.) it has a Gδ-dense set of transitive points.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. Realize that a point is transitive if and only if it visits all sets from a ∞ countable base {Ui }i=1 of open sets (note that X is compact). Thus

∞ ∞ ! \ [ −n tr(f ) = f (Uk ) . k=1 n=0 Now it is sufficient to use 1. ⇒ 8. in Theorem 1.2.

1.Transitive maps 1. Topologically Transitive Maps Transitive and intransitive points

First realize that in a dynamical system (X , f ), x ∈ intr(f ) ⇒ f (x) ∈ intr(f ) and f (x) ∈ tr(f ) ⇒ x ∈ tr(f ). If the system is without isolated points then we have equivalences instead of implications. Further, if a dynamical system (X , f ) is transitive then by Theorem 1.2 (1. ⇒ 13.) it has a Gδ-dense set of transitive points. We show how this can be proved (provided 1. ⇒ 8. has been established).

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps Transitive and intransitive points

First realize that in a dynamical system (X , f ), x ∈ intr(f ) ⇒ f (x) ∈ intr(f ) and f (x) ∈ tr(f ) ⇒ x ∈ tr(f ). If the system is without isolated points then we have equivalences instead of implications. Further, if a dynamical system (X , f ) is transitive then by Theorem 1.2 (1. ⇒ 13.) it has a Gδ-dense set of transitive points. We show how this can be proved (provided 1. ⇒ 8. has been established). Realize that a point is transitive if and only if it visits all sets from a ∞ countable base {Ui }i=1 of open sets (note that X is compact). Thus

∞ ∞ ! \ [ −n tr(f ) = f (Uk ) . k=1 n=0 Now it is sufficient to use 1. ⇒ 8. in Theorem 1.2.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. So, in an SDS (X , f ) there are the following possibilities: (a) tr(f ) = ∅, intr(f ) = X ,

(b) tr(f ) is dense Gδ and (b1) intr(f ) = ∅ (minimality) or (b2) intr(f ) is dense (Ex. 1.3).

1.Transitive maps 1. Topologically Transitive Maps Infinite systems without isolated points - SDS

Recall also that if in an SDS there is a transitive point then the system is transitive and hence the set of transitive points is dense Gδ.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps Infinite systems without isolated points - SDS

Recall also that if in an SDS there is a transitive point then the system is transitive and hence the set of transitive points is dense Gδ. So, in an SDS (X , f ) there are the following possibilities: (a) tr(f ) = ∅, intr(f ) = X ,

(b) tr(f ) is dense Gδ and (b1) intr(f ) = ∅ (minimality) or (b2) intr(f ) is dense (Ex. 1.3).

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps Infinite systems without isolated points - SDS

The following statement shows that no other possibility exists. Theorem 1.3 Let (X , f ) be an SDS. Then the set intr(f ) is either empty or dense in X (equivalently: if tr(f ) has nonempty interior then the system is minimal).

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps Infinite systems without isolated points - SDS

Beweis. Suppose that int(tr(f )) 6= ∅. This implies the transitivity of the system because it is standard. Since the preimage of a transitive point is a transitive point and the orbit of any transitive point intersects int(tr(f )), we have ∞ [ tr(f ) = f −n (int(tr(f ))) . n=0 Hence tr(f ) is open and, the system being transitive, dense. Then the set intr(f ) is closed and nowhere dense. Moreover, f (tr(f )) = tr(f ) and f (intr(f )) = intr(f ) (note that f is onto). We wish to prove that intr(f ) = ∅. Suppose on the contrary that this is not the case and take a closed neighbourhood U 6= X of the set intr(f ).

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps Infinite systems without isolated points - SDS

Beweis (Cont.) Then ∞ \ f −n(U) = intr(f ) n=0 since the orbit of any point from U \ intr(f ) intersects the open set X \ U. The set f (X \ int U) is compact and disjoint with intr(f ). So one can find in U a closed neighbourhood V of intr(f ) with f −1(V ) ⊂ U. Consequently,

∞ ∞ \ \ intr(f ) = f −n(V ) ⊂ f −n(U) = intr(f ) . n=1 n=0

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps Infinite systems without isolated points - SDS

Beweis (Cont.) Tn −k ∞ Denote Vn = k=1 f (V ). Then {Vn}n=1 is a decreasing sequence of T∞ closed sets with n=1 Vn = intr(f ) ⊂ int V . So there exists m such that

−1 −2 −m Vm = f (V ) ∩ f (V ) ∩ · · · ∩ f (V ) ⊂ int V .

Now define

−1 −(m−1) W = V ∩ Vm−1 = V ∩ f (V ) ∩ · · · ∩ f (V ) .

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps Infinite systems without isolated points - SDS

Beweis (Cont.) −1 −1 −2 −m Then W ⊂ V and f (W ) = f (V ) ∩ f (V ) ∩ · · · ∩ f (V ) = Vm = V ∩ Vm ⊂ V ∩ Vm−1 = W . Finally, realize that W is a closed neighbourhood of intr(f ). But the existence of a set W such that int(X \ W ) 6= ∅, int(W ) 6= ∅ and f −1(W ) ⊂ W contradicts the transitivity of f (the orbit of a point x ∈ tr(f ) ∩ (X \ W ) does not meet the set W ).

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps General references

1. S. Kolyada and L. Snoha, Some aspects of topological transitivity - a survey, Proc. ECIT-94, Grazer Mathematische Berichte, 334 (1997), 3-35. 2. S. Kolyada and L. Snoha, Topological transitivity, Scholarpedia, 4(2):5802 (2009).

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos. 1.Transitive maps 1. Topologically Transitive Maps HOMEWORK

Exercise 1.1 Let S be the unit circle and f : S → S be an irrational rotation. Then (S, f ) is topologically transitive, in fact minimal. Prove it.

Exercise 1.2 The map g : I → I , where I = [0, 1], defined by g(x) = 4x(1 − x) is topologically transitive. Prove it by using the fact that the tent map f (x) = 1 − |2x − 1| is topologically conjugate to g.

Sergiy Kolyada Topological Dynamics: Minimality, Entropy and Chaos.